REVEALED SOCIAL PREFERENCES ARTHUR DOLGOPOLOV MIKHAIL FREER

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1 REVEALED SOCIAL PREFERENCES ARTHUR DOLGOPOLOV Interdisciplinary Center for Economic Science, George Mason University MIKHAIL FREER ECARES, Université Libre de Bruxelles Abstract. We use a revealed preference approach to develop tests for the observed behavior to be consistent with theories of social preferences. In particular, we provide nonparametric criteria for the observed set of choices to be generated by inequality averse preferences and increasing benevolence preferences. These tests can be applied to games commonly used to study social preferences: dictator, ultimatum, investment (trust) and carrot-stick games. We further apply these tests to experimental data on dictator and ultimatum games. Finally, we show how to identify the levels of altruism and fair outcomes using the developed revealed preference conditions. Introduction Various studies show that people have other-regarding preferences (see e.g. Andreoni, 99; Andreoni and Miller, 22; Charness and Rabin, 22; Fisman, Kariv and Markovits, 27; Porter and Adams, 26; Castillo, Cross and Freer, 27). Moreover, other-regarding preferences are widely used in the applied theory (see e.g. Dufwenberg et al., 2; Maccheroni, Marinacci and Rustichini, 22; Szabo and Szolnoki, 22; Benjamin, 25). Significant research has been devoted towards understanding the motives for the social preferences (see e.g. Charness and Rabin, 22; Cox, 24). addresses: Dolgopolov: adolgopo@gmu.edu, Freer: mfreer@ulb.ac.be. Date: October 25, 28. We would like to thank Oyvind Aas, Christian Basteck, Laurens Cherchye, Bram De Rock, Thomas Demuynck, Daniel Houser, Iris Kesternich Georg Kirchsteiger and François Maniquet for insightful comments on the earlier versions of the paper.

2 2 A. DOLGOPOLOV AND M. FREER There are (at least) three most persistent motives: social welfare/efficiency, fairness and reciprocity. These four motives are covered by the two theories tested in this paper: inequality aversion and increasing benevolence. Inequality aversion assumes that players extract utility from their own payoffs and encounters some disutility if payoffs are unbalanced (see e.g. Fehr and Schmidt, 999; Bolton and Ockenfels, 2). Hence, inequality aversion, encompasses the fairness motive. Increasing benevolence assumes that a player s willingness to pay for an additional dollar received by another player increases in the player s own payoff (see Cox, Friedman and Sadiraj, 28). Underlying intuition implies that the payoff of another player is a normal good. Hence, increasing benevolence encompasses reciprocity motive and the social welfare considerations if these are present. Finally, let us note, that both theories are widely used in applied theory research. The paper constructs revealed preference tests for inequality averse and increasing benevolence preferences. We start with a standard choice environment over linear budgets in which a player decides how to allocate money between herself and another player (dictator game). Next, we generalize tests to other games commonly used to study social preferences: ultimatum, trust, and carrot-stick games. Note that social welfare considerations are not prevalent in these applications. In addition, we apply the tests to the experimental data on dictator and ultimatum games. There are three main empirical findings. First, both theories explain behavior better than random decision-making. Second, both inequality averse and increasing benevolence preferences are strictly nested within the other-regarding preferences. That is there is a significant share of population consistent with having other-regarding preferences, but not consistent with either of nested theories. Third, inequality aversion explains the behavior of subjects quite well in both dictator (better than increasing benevolence) and ultimatum game. However, the degree to which inequality aversion prevails over increasing benevolence different for different populations. The revealed preference approach, pioneered by Samuelson (938), originates in the fact that, we can only observe the choices of players but not their preference relations. Inequality aversion is popular in political economy (see e.g. Fong, 2; Tyran and Sausgruber, 26; Höchtl, Sausgruber and Tyran, 22; Durante, Putterman and Van der Weele, 24; Agranov and Palfrey, 25). Increasing benevolence has been used by Cox, Friedman and Sadiraj (28) to model the behavior in two-stage games and by (Benjamin, 25) to guarantee the efficiency in the bilateral exchange with social preferences.

3 REVEALED SOCIAL PREFERENCES 3 Revealed preference theory allows avoiding functional misspecification of preferences. Starting with Richter (966) and Afriat (973) the approach has been applied to construct tests of individual and collective decision-making (see Chambers and Echenique, 26, for a comprehensive overview of the results). Revealed preference theory has been applied to other-regarding preferences starting with Andreoni and Miller (22). Next, we provide a connection to the previous results on revealed social preferences. Cox, Friedman and Sadiraj (28) provides necessary revealed preferences conditions for observed choices to be consistent with increasing benevolence and provides a method of comparing subjects in terms of altruism if the demand functions are completely observed. We show that conditions proposed are also sufficient and that the comparisons in terms of altruism can be applied even if only the finite set of choices is observed. 2 Deb, Gazzale and Kotchen (24) constructs revealed preference tests for a special case of inequality aversion (inequality aversion in differences) if budgets are linear. The test we construct does not depend on the specification of the inequality measure. That is, if a player is consistent with inequality averse preferences, she is consistent with inequality aversion preferences given any measure of inequality. Moreover, the test does not require linearity of budgets and therefore, has a larger scope of applications. The remainder of this paper is organized as follows. Section 2 presents the general set up and the revealed preference tests as well as extensions of the test for other games. Section 3 provides empirical illustrations. Section 4 presents the partial identification of the level of altruism and notion of fair outcome using the revealed preference conditions. Section 5 provides concluding remarks. All proofs are collected in Appendix A. 2 Theoretical Framework We consider a dictator game, which is structured as follows. A player decides how to allocate a given amount of money between herself and the other player, and the chosen allocation is implemented. This game can be written as a decision problem, in which one player chooses a two-dimensional vector allocation: payoff to self and payoff to another player. Let X R 2 + be the set of alternatives. For every x X let x = (x s, x o ), where x s is the payoff to self and x o is the payoff to another player. Let p R 2 ++ be a price 2 The original paper compared the altruism levels either pointwise or if the entire demand function is observed. However, we preserve the assumption of the Cox, Friedman and Sadiraj (28) that two players had to face the same experiment (making choices over similar budgets).

4 4 A. DOLGOPOLOV AND M. FREER vector. Income is normalized to one at every point, and the budget set is defined as, B(p) = {x X : px }. Let E = (x t, p t ) T t= be an experiment, which consists of T choices (x t ) at a given price vector (p t ). Moreover, we assume that chosen points x t are such that p t x t =. 3 A function u(x) : X R rationalizes the consumption experiment E if for all y B(p t ), u(x t ) u(y) for every t {,..., T }. In what follows we present revealed preference tests for each of the theories of social preferences. We start with other-regarding preferences. Next, we present the test for inequality averse preferences and the test for increasing benevolence preferences. Next, we show that the latter two theories are independent. Finally, we show how to apply the tests to ultimatum, trust and carrot-stick games. 2. Other-Regarding Preferences (OR). Other-regarding preferences assume that a player cares about her own payoff and the payoff of the recipient. Theory does not make an explicit assumption of whether the player derives utility or disutility from x o. Definition. An experiment E = (x t, p t ) T t= is rationalizable with other-regarding preferences if there is a continuous and locally non-satiated utility function u(x s, x o ) that rationalizes E. Other-regarding preferences include utility function that is monotone in both payoffs (altruistic preferences) as a special case. Hence, rationalizability with other-regarding preferences can be deduced to the regular case of existence of locally non-satiated utility function over two-dimensional space of real outcomes. Definition 2. An experiment E = (x t, p t ) T t= is consistent with Generalized Axiom of Revealed Preference (GARP) if and only if we have p t x tn p t x t for all sequences x t,..., x tn, such that p t j+ x t j p t j+ x t j+, j {i,..., n }. Figure presents the violation of GARP. An allocation x is chosen given prices p, therefore, it is better than any allocation which is available at p. At the same time x 2 is available at p, therefore, x is strictly directly revealed preferred to x 2. Finally x 2 is directly revealed preferred to x, since x is available at p 2. 4 Hence, observed choices could not be generated by maximization of utility function. 3 This technical assumption is dictated by non-satiation of preferences. All the further reasoning can be done without this assumption using more complicated notation. 4 Formally, this is a violation of Weak Axiom of Revealed Preferences (WARP) and in the case of two-dimensional linear budgets WARP is equivalent to GARP (see e.g. Rose, 958). However, we

5 REVEALED SOCIAL PREFERENCES 5 x o x x 2 p p 2 x s Figure. Other-Regarding Preferences and GARP Proposition (Afriat (967); Diewert (973); Varian (982)). An experiment is rationalizable with other-regarding preferences if and only if it satisfies GARP. 2.2 Inequality Aversion (IA) Inequality aversion assumes that player gets utility from her own payoff and disutility if payoffs are unbalanced, In order to quantify the unbalancedness of the payoffs we use the inequality measure. commonly used inequality measures are presented below. Some examples of Inequality in differences (e.g. Fehr and Schmidt, 999; Tyran and Sausgruber, 26; Agranov and Palfrey, 25): x s x o f(x s, x o ) = β(x s x o ) where β. if x s x o if x o > x s Inequality in shares or Lorenz curve 5 (e.g., Bolton and Ockenfels, 2): x s x f(x s, x o ) = s+x o if x 2 s x o ( ) β if x o > x s where β. Gini Index f(x s, x o ) = xs xo 2(x s+x o) 24) x s x s+x o 2 (e.g. Durante, Putterman and Van der Weele, prefer to introduce the GARP, since further we deal with nonlinear budgets for which the result does not necessarily hold. 5 Intuition for Lorenz curve is the same as for inequality in shares for the two-player case since the shape of the curve is determined by deviation of the lower payoff from equal share split.

6 6 A. DOLGOPOLOV AND M. FREER Further we present an axiomatization of the inequality measure. Closely related axiomatization has been used by Fehr, Kirchsteiger and Riedl (998) for the gift exchange game. This axiomatization generalizes the examples of inequality measures presented above. Definition 3. A continuous function f(x s, x o ) is an inequality measure if: f(x s, x o ), for every x s, x o ; f(x s, x o ) = if and only if x s = x o ; if x s > x o, then f(x s, x o ) is decreasing in x o and increasing in x s ; if x s < x o, then f(x s, x o ) is increasing in x o and decreasing in x s ; f(max{x s, x o }, min{x s, x o }) f(min{x s, x o }, max{x s, x o }). Further we present the definition for rationalization with inequality averse preferences. This rationalization, in general, depends on the inequality measure chosen. Definition 4. Let f(x s, x o ) be an inequality measure. An experiment is rationalizable with inequality averse preferences if there is a continuous utility function u(x s, f(x s, x o )) increasing in x s and decreasing in f(x s, x o ) that rationalizes it. Rationalizability with inequality averse preferences requires every player to choose to allocate to herself at least as much as to the other player. This condition is necessary, because if x s < x o, then the player could obtain greater utility by increasing x s at the cost of x o. Hence, player can set up x s > x s and x o < x o such that x s x o. In this case f(x s, x o) < f(x s, x o ) and therefore, (x s, x o) should be strictly better than (x s, x o ). That is the player has chosen a strictly dominated outcome and therefore cannot be rationalized with maximization of inequality averse utility function. This condition together with GARP is sufficient for rationalizability with inequality averse preferences. Proposition 2. Let f(x s, x o ) be an inequality measure. An experiment is rationalizable with inequality averse preferences if and only if it satisfies GARP and x t s x t o for every t {,..., T }. 6 6 The proposition is presented assuming budgets to be linear, while the proof is provided for nonlinear budgets as well. This is done to avoid further abuse of notation in the main text and state the result which can be applied to dictator as well as to other games studied further.

7 REVEALED SOCIAL PREFERENCES 7 Condition for rationalization with inequality averse preferences does not depend on the inequality measure. That is, condition is the same for any inequality measure. Hence, the following corollary immediately follows from Proposition 2. Corollary. An experiment is rationalizable with inequality averse preferences, if and only if it is rationalizable with any inequality measure. The implications of Corollary are two-fold. On one hand, we cannot exploit the particular structure of the inequality measure to refine the test. On another hand, we can test the comprehensive assumption of inequality aversion, which does not depend on the particular form of the inequality measure to be assumed. 2.3 Increasing Benevolence (IB) Increasing benevolence means that a player s willingness to pay for an additional dollar given to the other player is increasing in own payoff x s. 7 Unlike in the preiovous cases we use the statement in terms of the demand functions. Denote the demand function for x o by D o (p s, p o ) and the demand for x s by D s (p s, p o ). Since we operate in a two-dimensional case, one demand can be immediately derived from another D s (p s, p o ) = podo(ps,po) p s. Definition 5. An experiment is rationalizable with increasing benevolence preferences if there is a rational demand function D o (p s, p o ) such that D o (p t s, p t o) = x t o, and po p s p o p s and p o Do(ps,po) p s D s (p s, p o ) implies D o (p s, p o ) D o (p s, p o). Increasing benevolence is equivalent to normality of x o. A good is said to be normal if its demand is increasing function of income. Necessity of normality for increasing benevolence is quite obvious. Figure 2 illustrates why normality is sufficient for increasing benevolence. Assume that x is a point chosen from the budget defined by p ; then, the new budget is such that ps p o p s (x p s is relatively more expensive in the new o budget) and the old bundle is attainable. The dashed line shows the parallel downward shift of the budget defined by p 2. Hence, the choice from the dashed budget should be 7 This can be defined more formally with the marginal rate of substitution W T P = /MRS = uxo u xs is increasing in x s. We use the reduced form definition of this, which is necessary but not sufficient. However, it is sufficient to guarantee the empirical implications described by Cox, Friedman and Sadiraj (28). Moreover, if we define M RS via the ratio of the inverse demand functions (to guarantee the existence of M RS), some sufficiency result can be inferred. Although, one can easily check that if we, for instance, assume that x s and x o are substitutes, then the demand conditions would be sufficient for the MRS version.

8 8 A. DOLGOPOLOV AND M. FREER x o x p p 2 x s Figure 2. Increasing Benevolence and Normality with at least as much x o as from p (due to the substitution effect). Furthermore, since dashed and p 2 budgets are different only in income, then normality would guarantee that the choice from p 2 would be above the x. Definition 6. An experiment E = (x t, p t ) T t= is consistent with Normality Axiom of Revealed Preference (NARP) if and only if for all observations t, v {,..., T } if p t o/p t s p v o/p v s and x v s pt o xv o, then x v p t o x t o. s Equivalence between increasing benevolence and normality of demand in x o allows us to employ the result from Cherchye, Demuynck and De Rock (28) as the test for increasing benevolence. 8 Proposition 3 (Cherchye, Demuynck and De Rock (28)). An experiment is rationalizable with increasing benevolence preferences if and only if it satisfies NARP. 2.4 Independence of Nested Theories Further we show that nested theories (inequality aversion and increasing benevolence) are independent. Moreover, they are not exhaustive there can be an other-regarding preference relation, neither inequality averse nor increasing benevolent. Hence, there are four cases: preferences consistent with both nested theories, with only one of them or with neither of them. Next we provide examples and intuition for each case. 8 If a reader is not convinced by the equivalence argument above, please see p.375 in Cherchye, Demuynck and De Rock (28) where it is directly proven that the increasing benevolence property is satisfied.

9 REVEALED SOCIAL PREFERENCES 9 x o x o p 2 x 2 x p x 2 x s x p p 2 x s (a) IA but not IB (b) IB but not IA x o x o p 2 p 2 x p p x 2 x x 2 x s x s (c) IA and IB (d) OR, but neither IA nor IB Figure 3. Independence of Inequality Aversion and Increasing Benevolence Consider budgets from Figure 3. Inequality aversion predicts choices to be at or below the 45 degree line from the origin. Increasing benevolence requires the choice from p 2 to have more x o than the choice from p. Figure 3(a) presents the case for the preferences to be consistent with inequality averse preferences, but not increasing benevolent. Indifference curves presented could be generated by the utility function u(x s, x o ) = x 3 s max( x s x o, ). It guarantees that for p the optimal point is an allocation close to equal split, while for p 2, the optimal choice is to spend all income on x s. Therefore, this is a violation of NARP and choices are not consistent with increasing benevolence preferences. Figure 3(b) presents the case of preferences that are increasing benevolent, but not inequality averse. Assume that player maximizes the utility function u(x s, x o ) = x s x 2 o. Then, for hight enough incomes (and low enough p o ) the choice would lie above 45 degree line (x s < x o ). Hence, such preferences would not be consistent with inequality aversion. Figure 3(c) presents example of

10 A. DOLGOPOLOV AND M. FREER preferences consistent with both theories. Examples of such preference relations include selfish preferences (u(x s, x o ) = x s, presented on the figure) and perfect complements (u(x s, x o ) = min(αx s, x o ) with α ). Finally, Figure 3 presents example of otherregarding preferences not consistent with either of nested theories. Example of such preference is u(x s, x o ) = 4 max(x s, 3) + x o. Idea behind, is that for low income, player only cares about x o and as soon as she gets enough income, she becomes more selfish. Therefore, choice from p would be above 45 degree line at the same time in the budget p 2 player would choose less of x o then under p. 2.5 Revealed Social Preferences Beyond Dictator Games Extending the theory of revealed social preferences to other games is of particular importance, because different motives (that can depend on the game) can trigger different theories to perform better (see for instance Engellman and Strobel, 24). Further we show how the revealed preference tests of social preferences can be applied for ultimatum, investment and carrot-stick games. Following Cox, Friedman and Sadiraj (28) we consider second-movers in the two-stage games Ultimatum Game. First-mover is given an endowment m t and asked to allocate it between herself and a second-mover, given that p t ox o + p t sx s = m t, where x o denotes the first-mover s earnings and x s denotes the second-mover s earnings. Recall that we analyze the game from the point of view of the second-mover, therefore, payoff to proposer is considered to be as x o and payoff to responder as x s. Second-mover decides whether to accept or reject the proposed allocation. If the allocation is accepted, it is implemented; otherwise, both players get zero. x s A t (f(x t s, x t o), x t s) R t f(x s, x o ) Figure 4. Acceptance and Rejection regions in Ultimatum Game

11 REVEALED SOCIAL PREFERENCES The case of other-regarding preferences is already considered in Castillo, Cross and Freer (27). Increasing benevolence is not well-defined for the binary choices. Hence, we are left to test for inequality aversion. The experiment, on the side of the responder, is a sequence of binary decisions between proposed allocations and zero payoffs. Figure 4 presents the decision problem, as well as acceptance (A t ) and rejection (R t ) regions in (f(x s, x o ), x s ) coordinates. To be more precise, acceptance and rejection regions can be defined as follows. A t = {(f(x s, x o ), x s ) : x s x t s and f(x s, x o ) f(x t s, x t o)} and R t = {(f(x s, x o ), x s ) : x s x t s and f(x s, x o ) f(x t s, x t o)} If point y t = (f(x t s, x t o), x t s) was accepted, then it is revealed preferred to zero. Hence, every point in which is strictly better than y t should also be better than zero (by transitivity), thereofre, should be accepted. If y t is rejected, then zero is revealed better than y t. Hence, every point which is strictly worse than y t is also strictly worse than zero (by transitivity), therefore, should be rejected. Note that acceptance and rejecting regions correspond to the better than and worse than sets imputed from the partial order imposed by inequality aversion. Hence, if y t is accepted, then every point from its acceptance region should be accepted and if y t is rejected, then every points from its rejecting region should be rejected. Denote the set of all accepted allocations by A x and the set of all rejected allocations by R x. Corollary 2. Let f(x s, x o ) be a measure of inequality. An ultimatum game experiment is rationalizable with inequality averse preferences if and only if ( ) R x = x t A x A t and ( ) A x =. x t R x R t Acceptance and rejection regions translates for different measures of inequality. Hence, performance of different measures of inequality can be distinguished. To illustrate this, we consider examples of inequality aversion in differences (f(x s, x o ) = x x s x o ) and inequality aversion in shares (f(x s, x o ) = s ). We choose this x s+x o 2

12 2 A. DOLGOPOLOV AND M. FREER x s Slope of Slope of x s A t A t R t R t x o x o (a) in Differences (b) in Shares Figure 5. Inequality Aversion in Ultimatum Game for Different Measures of Inequality (x s > x o ) measures for the matter of illustration, since they are among the most widely applied in the literature. Figure 5(a) shows the acceptance and rejection regions for the inequality aversion in difference. Every point on the line of slope one which goes through x has the same inequality level as x. Hence, the area between two dashed lines of slope one delivers the inequality level which is less or equal than inequality level at x. Therefore, the area between these lines and above the horizontal dashed line (current level of x s ) is strictly better than x and specifies the acceptance region. Rejection regions (shaded regions below the horizontal line) give the second-mover lower payoff and increase inequality, therefore, is strictly worse than x according to the inequality averse partial order. Figure 5(b) shows acceptance and rejection regions for inequality aversion in shares. Every point on the line that goes through zero and x has the same inequality level as x. The acceptance and rejection regions constructed by exactly the same logic as on Figure 5(a), although with different lines that preserve inequality. Comparing rejection and acceptance regions from Figures 5(a) and 5(b), we can see that they are different. Therefore, these measures of inequality have different testable implications Investment Game. Players start with an endowment of I. The first-mover sends an amount s [, I] to the second-mover who receives ks, for k >. Then the second-mover returns an amount of r [, ks], and the first-mover receives pr, where p k. The final payoffs are x o = I s + pr and x s = I + ks r for the first- and

13 REVEALED SOCIAL PREFERENCES 3 second-mover correspondingly. Hence, a family of investment games with different p and different s sufficient price generates variation to apply revealed preference tests. x o x o B t B s B t x s x s (a) Violation of GARP (b) Downward closure of budgets. Figure 6. Second-mover s budget set in the investment game. Figure 6(a) presents the budget set of the second-mover and the possible violation of GARP in this case. Choices on the horizontal segments are not feasible. However, to construct the precise test we need to take the downward closures of the budget sets presented at the Figure 6(b). Denote by. B = {y : there is x [, ks] such that x y} the downward closure of B and by B the interior of the downward closure (replacing the weak inequality with strict one). Assume k to be fixed over the observations, hence, investment game experiment consists of observed triples of s t (determines the income), p t (relative price of returning) and x t (chosen point). Hence, we can restate GARP using x B instead of px m and x B instead of px < m. Using the new notation Proposition can be immediately applied (using the Forges and Minelli, 29, result) therefore, the proof is omitted. 9 9 Formally we also have to assume that burning money is feasible, i.e. players can choose in the interior of the budget set. Although, the necessity of GARP holds for both altruistic (player gets utility from both x s and x o ) and spiteful (player gets utility from own payoff and disutility from x o ) preferences, while sufficiency can be inferred from Nishimura, Ok and Quah (27) result. Idea behind is that spiteful player will always choose to return nothing and this choice pattern is consistent with GARP.

14 4 A. DOLGOPOLOV AND M. FREER Corollary 3. An investment game experiment E = (x t, B t ) T t= is rationalizable with other-regarding preferences if and only it satisfies GARP. Next, we consider increasing benevolence preferences. If all choices are such that x t B v if and only if p v x t s t for every t, v {,..., T }, then Proposition 3 can be applied to test for increasing benevolence. This assumption reduces the budgets to the linear case of dictator game. Considering the inequality aversion, we can immediately apply the Proposition 2, substituting the linear definition of the budget with the general definition. The reason behind is that since s t, then x s = x o is available even for s = and clearly available for every s >. Corollary 4. Let f(x s, x o ) be an inequality measure. An investment game experiment E = (x t, B t ) T t= is rationalizable with inequality averse preferences if and only if it satisfes GARP and x t s x t o for every t {,..., T }. Moreover, if an investment game experiment experiment is rationalizable with an inequality measure, then it is rationalizable with any inequality measure Carrot-Stick Game. Both players start with an endowment of I. The firstmover chooses the amount to be sent s [, I]. Then, the second-mover can return the amount r [ ks, ks] and the first-mover receives pr. The final payoffs are x o = I s + pr and x s = I + ks r. Hence, the family of carrot-stick games with different different p and different s generates sufficient price variation. Carrot-stick experiment is defined by s t, p t and x t. x o x o x o B t B s x s x s x s (a) B (b) B (c) B Figure 7. Second-mover s budget set in the carrot-stick game. Figure 7(a) presents the budget sets that the second-mover faces. Unlike in the cases of dictator and investment games there implications are different for altruistic

15 REVEALED SOCIAL PREFERENCES 5 and spiteful preferences, because there are both upper and lower borders of the budgets. Preferences are said to be altruistic if utility is increasing in both x s and x o. Preferences are said to be spiteful if utility is increasing in x s and decreasing in x o. Therefore, the altruistic player use only carrot, while spiteful use only stick, and each of the choices is dominated for the different type of player. Denote by B t the budget (as on Figure 7) based on which the player makes a choice. Denote by B [ ] = {(x s, x o) : (x s, x o ) B such that x s x s and x o ( )x o }. Denote by B [ ] = {(x s, x o) : (x s, x o ) B such that x s x s and x o ( )x o } with at least one inequality being strict. Figures 7(b) and 7(c) illustrate the construction of B and B respectively. The shaded areas show the part of the space added by taking the closure of the budget. Further we refer to the A-GARP as to GARP which uses x B instead of px and x B instead of px < and to S-GARP as to GARP which uses x B instead of px and x B instead of px <. We als claim that a choice is a violation of A-GARP is it is in B and a choiceis a violation of S-GARP if it is in B. Hence, we can immediately provide the criteria for rationalization with altruistic and spiteful preferences using the result from Nishimura, Ok and Quah (27). Corollary 5. A carrot-stick experiment E = (x t, B t ) T t= is rationalizable with altruistic [spiteful] preferences if and only if it satisfies A-GARP [S-GARP]. Denote by p t price vector, that corresponds to the upper boundary of B t. If all choices are such that x t B v if and only if p v x t for every t, v {,..., T }, then Proposition 3 can be applied to test for increasing benevolence. Recall that increasing benevolence is nested within altruistic preferences. Therefore, the stick is not consistent with increasing benevolent preferences. Next we consider inequality-averse preferences. As in ultimatum game, in the carrotstick game different measures of inequality have different empirical implications. Figure 8(a) presents the case with an inequality averse (in differences) player using the stick. The shaded area presents the set of points better than the chosen action. None of the points that dominate the chosen one are in the budget. Hence, the choice of stick can be optimal if a player has inequality averse preferences. Figure 8(b) shows that if p is low enough, then using stick is no longer rational. Figure 8(c) shows that The test given the measure of inequality can be easily formulated following the result of Nishimura, Ok and Quah (27) as we have done for the case of ultimatum game.

16 6 A. DOLGOPOLOV AND M. FREER x o x o x o x s x s x s (a) (b) (c) Figure 8. Using the stick with inequality averse preferences. testable implications in the carrot-stick game would depend on the particular measure of inequality, because the budget set contains the points that dominate the chosen one. Hence, the same choice is consistent with inequality aversion in differences (Figure 8(a)), but not with inequality aversion in shares (Figure 8(c)). Note that inequality aversion is the only of the above-mentioned theories that can rationalize a player who uses both carrot and stick. Further we present restrictions on the experimental design which allows us to apply test for inequality aversion without making parametric restrictions about the measure of inequality. Idea behind is that we need to restrict the experiment to the collection of budgets such that x s = x o is available. In order to implement this, it is enough to either guarantee the second-mover the initial endowment of I. The same logic as for investment game implies that for every s, the equal outcome is available. Alternatively, one can restrict the minimal amount sent by the first-mover to s I to k+ guarantee that equal outcome is available. Hence, in this case the version of Proposition 2 can be applied to test for rationalizability with inequality aversion (using the altruistic version of GARP from Corollary 5). Finally, in this case inequality averse player would use carrot only and never use a stick. Corollary 6. Let f(x s, x o ) be an inequality measure and E = (x t, B t ) T t= be carrot-stick experiment such that x s = x o outcome is available at every budget set. A carrot-stick experiment E = (x t, B t ) T t= is rationalizable with inequality averse preferences if and only if it satisfies A-GARP and x t s x t o for every t {,..., T }. Moreover, if a carrot-stick experiment is rationalizable with an inequality measure, then it is rationalizable with any inequality measure.

17 REVEALED SOCIAL PREFERENCES 7 Let us conclude with a remark on possible experimental design in order to apply the tests. Design of experiment directly follows from the construction of the test. In order to guarantee the sufficient price variation one can use discretized first-mover s problem with a strategy method. Hence, the second-mover would have to make decisions over the budgets corresponding to every possible decision (out of the finite set) the firstmover can make for every budget proposer would face (see Castillo and Cross, 28; Castillo, Cross and Freer, 27, for the case of ultimatum game). Strategy method allows to avoid possibility of falling short on the power of test due to specific decisions first mover made. 3 Empirical Illustration We present evidence from dictator and ultimatum games. While dictator game allows for comprehensive test of inequality aversion hypothesis (see Corollary ), ultimatum game (theoretically) allows to distinguish between different measures of inequality. 3. Dictator Game We use data from two studies of dictator games. In both studies, subjects repeatedly played a dictator game with different relative prices and endowments. In every period subjects were asked to allocate tokens between themselves and another person, choosing a point on a linear budget p t sx s + p t ox o m t. The first study (Fisman, Kariv and Markovits, 27) contains results of experiments with 76 undergraduates from UC Berkeley. In this study, every subject faced 5 different budgets with randomly determined prices. The second study (Porter and Adams, 26) contains results of experiments with 89 subjects recruited from the general population from the southeast region of the UK. In this study every subject faced different budgets with predetermined prices. 3.. Consistency Results When applied to data, notions of rationality prove to be very strict at least for the first data set: no more than 6% of subjects can be rationalized with other-regarding preferences and no more than % with nested theories. Therefore, it makes sense to relax the notion of rationality and allow for some probability that people make mistakes. For this purpose, we use the Houtman-Maks index (HMI). 2 HMI is the maximum fraction of data that can be rationalized by a given Experiment contains two other treatments which we do not consider in our analysis. One of the treatments uses step-shaped budgets and another is a dictator game with two recipients. 2 See Houtman and Maks (985); Heufer and Hjertstrand (25); Dean and Martin (29). We use the HMI because it is the only index that can be applied to test Inequality Aversion. Critical Cost

18 8 A. DOLGOPOLOV AND M. FREER theory. That is, if in a total of T observations, the maximum subset which is consistent with the theory is τ, then HMI = τ/t. For the technical details regarding the implementation of the HMI index, see Supplementary Materials. We report results for the HMI level of.9. The results are robust to other levels of HMI (see Supplementary Materials). The HMI level of.9 allows for deviations from rationality in no more than % of budgets. Next, we want to control for false positives. A false positive is the probability that a random decision making would look consistent with the test. We use two procedures which differ in the assumption about the random behavior. First is the Bronars (987) power, conducted by generating random subjects who make decisions uniformly distributed along the budget line. Power of the test is computed as the fraction of random subjects who fail to perform consistently with the test. The second is the bootstrap power (see e.g. Cox, 997; Harbaugh, Krause and Berry, 2; Andreoni and Miller, 22). This measure controls for possible behavioral rules that can cause false positive results even if people would take decisions at random. To compute the bootstrap power of the test, we calculate the empirical distribution of the shares of income spent on each commodity in our case the subject s payoff and the other s payoff and simulate the pseudo subjects who make their choices at random but distributed according to the empirical distribution function. Last, to compare pass rates controlling for the power we use the predictive success index (PSI) introduced by Selten (99). 3 The predictive success index is defined as the difference between the share of people that satisfies an axiom at the given level of HMI and the probability that random choices will satisfy the axiom at the same level of HMI. This index ranges between and, with meaning no subject passes while all random subjects pass and meaning every subject passes while none of the random subjects do. If PSI is greater than zero, then theory describes the behavior better than random choice, and if PSI is less or equal to zero, then the random choice explains the observed behavior better. Efficiency Index introduced by Afriat (973) would not adequately work in the context of inequality aversion. The money pump index introduced by Echenique, Lee and Shum (2) is defined for GARP only. The swaps index proposed by Apesteguia and Ballester (25) can be applied only in the context of finite choice sets. 3 Methodology of using predictive success index in the revealed preference context was introduced by Beatty and Crawford (2). Statistical interpretation of the index which allows us to construct confidence intervals was proposed by Demuynck (25).

19 REVEALED SOCIAL PREFERENCES 9 Table shows results of testing other-regarding preferences for both datasets. The second column presents the pass rates (share of subjects who pass the test with HMI at least.9). The third and fourth columns present the power computations according to Bronars and the bootstrap methods. 4 Last two columns present the predictive success index using Bronars and bootstrap powers. Power of Test PSI Theory Pass Rate Bronars Bootstrap Bronars Bootstrap Fisman, Kariv and Markovits (27) data Other-regarding 58 (76.32%).% 99.63% % conf. interval (65.8% %) (99.99% -.%) (99.58% %) ( ) ( ) Porter and Adams (26) data Other-regarding 8 (9.%) 68.72% 8.% % conf. interval (83.5% %) (68.4% %) (79.75% - 88%) ( ) ( ) Table. Results for Other-Regarding Preferences Subjects (in both experiments) are consistent with having other-regarding preferences. In particular 76% of subjects in the first dataset and 9% in the second one are consistent with having other-regarding preferences. Results are robust to controlling for the power of test. Inequality aversion and increasing benevolence are nested within the other-regarding preferences model. That is, a subject can only be consistent with having inequality averse and/or increasing benevolence preferences, if she is consistent with having otherregarding preferences. Hence, we report a nested theory analysis; that is, results are presented for the subexperiment, which consists only of subjects who are consistent with other-regarding preferences hypothesis (given HMI=.9). Table 2 presents results for nested theory analysis. Structure of the table is the same to the one of Table. Both theories are significantly restrictive there is significant share of population (5-79%) which is consistent with having other-regarding preferences, but not with increasing benevolence or inequality averse preferences. In addition, while 55% of subjects are consistent with inequality averse preferences in 4 Power is different for different data sets first of all, because of the different amount of budgets: 5 vs. Difference in power is even larger for increasing benevolence. Experiment of Porter and Adams (26) is aimed at testing GARP, which rather requires price variations while NARP requires rather income variation. Income variation is higher in Fisman, Kariv and Markovits (27) data set because of the larger amount of budgets.

20 2 A. DOLGOPOLOV AND M. FREER Power of Test PSI Theory Pass Rate Bronars Bootstrap Bronars Bootstrap Fisman, Kariv and Markovits (27) data Inequality Aversion 32 (55.7%).% 92.58% % conf. interval (4.54% %) (.% -.%) (92.39% %) (2 -.68) ( ) Increasing Benevolence 2 (2.69%).%.% 95% conf. interval (.7% %) (.% -.%) (.% -.%) (. -.3) (. -.3) Porter and Adams (26) data Inequality Aversion 5 (62.96%) 98.62% 9.5% % conf. interval (5.5% %) (98.55% %) (9.3% %) ( ) (3 -.64) Increasing Benevolence 69 (85.9%) 85.68% 65.67% % conf. interval (75.55% - 92.%) (85.44% %) (65.35% %) ( ) (3 -.59) Table 2. Results for Nested Theories Fisman, Kariv and Markovits (27) data, only 2% of them is consistent with having increasing benevolence preferences. However, the results are opposite for the Porter and Adams (26) data: 63% of subjects are consistent with having inequality averse preferences and 85% of them are consistent with having increasing benevolence preferences. To sum up, inequality aversion appears to describe data better in the first dataset; and increasing benevolence performs at least as well as inequality aversion in the second one (difference in PSIs is not statistically significant). This inconsistency between the two datasets provides evidence in line with Fehr, Naef and Schmidt (26), who showed that social preferences may depend on the demographic characteristics of the population Mixed Types Analysis None of the nested theories can explain the behavior of the entire sample. At the same time, both theories perform well even conditioning on their power. In addition, different theories have quite different empirical implications, and the correlations between pass rates for inequality aversion and increasing benevolence are quite low:.22 (with a confidence interval of [.,.42]) for Fisman, Kariv and Markovits (27) data and.5 (with a confidence interval of [.34,.65]) for Porter and Adams (26) data. 5 This shows that there is a non-trivial probability that different 5 Given that the tests are binary, appropriate statistic is φ-coefficient. It is a version of correlation coefficient for two binary variables. Both logit and probit regression coefficients are insignificant for Fisman, Kariv and Markovits (27) data: logit regression coefficient is.2 (with 95% confidence interval of [.6, 2.6]) and probit regression coefficient is.75 (with 95% confidence interval of [.4,.57]). That is we can not reject that two nested theories are unrelated for Fisman, Kariv and Markovits (27) data. For Porter and Adams (26) data the relationship is much stronger: logit coefficient is 3.9 (with 95% confidence interval of [.74, 4.99]), probit coefficient is.84 (with

21 REVEALED SOCIAL PREFERENCES 2 subjects can be consistent with different notions of rationality. Therefore, we perform a mixed type analysis. Main goal of this exercise is to find out which theory is the most appropriate to describe behavior at the subject level, while the previous analysis focused at the sample level. Y Y IA or IB (4%) 49(55%) HMI IB α IB Y HMI OR α OR HMI IA α IA N N Y HMI IB α IB IA IB 2(28%) 2(2%) 7(9%) 7(9%) N IC 8(24%) 8(9%) N OR 9(25%) (%) Top numbers are for Fisman, Kariv and Markovits (27) data, bottom numbers are for Porter and Adams (26) data. Figure 9. Classification Tree for Dictator Game Subjects are assigned to theories according to three sequential binary classification steps presented in Figure 9. First, if a subject is not consistent with other-regarding preferences at threshold α OR, she is classified as inconsistent with other-regarding preferences (IC). Next, we compare whether she is consistent with inequality aversion or increasing benevolence with thresholds α IA and α IB respectively. If the subject is not consistent with either, she is classified as other-regarding (OR). If the subject is consistent with both, she is assigned to a separate class of inequality averse or increasing benevolent (IA or IB). If the subject is consistent with only one theory, she is classified as inequality averse (IA) or increasing benevolent (IB). It is still necessary to determine the thresholds for the classification tree. In order to do this, we modify the unsupervised machine learning methodology from Liu, Xia and Yu (2). The approach maximizes the information gain from adding a particular cluster. We base this measure on HMI, but the approach is general (for more detailed explanation see Supplementary Materials). The thresholds obtained are as follows: 95% confidence interval of [.7, 2.72]). That is, odds of the subject being consistent with inequality aversion are at least e.74 = 5.7 times higher if she is also consistent with increasing benevolence than if she is not.

22 22 A. DOLGOPOLOV AND M. FREER α OR = 45/5; α IA = α IB = 4/5 for Fisman, Kariv and Markovits (27) data and α OR = α IA = α IB = / for Porter and Adams (26) data. In the first dataset, a large set of subjects can be described by inequality aversion but not by increasing benevolence preferences (28% against 9%). In the second experiment, the distinction is less clear, as the majority of subject (55%) can be described by IA or IB. 6 Remark that, -25% of the population cannot be explained by either inequality aversion or increasing benevolence, but still has other-regarding preferences. This fact also provides additional evidence for both assumptions being significantly restrictive. The difference between the two datasets in terms of classification for nested theories is consistent with the results in the previous subsection. Given the significant share of population consistent with both: IA and IB especially in Porter and Adams (26) data set, we conduct the cross power analysis. That is we need to simulate the random subjects which are consistent with IA but not with IB and vise versa. This allows us to construct the modified predictive success given the conditional power analysis. Using the results of cross power analysis we can construct the adjusted predictive success index (APSI). To explain the idea of APSI we provide an example of its computation for IA. We consider separate PSIs for the subsamples of IA or IB and IA. The main difference is that for IA or IB subsample we use the cross power. APSI in its order is equal to the weighted sum of the PSIs for each subsample, where weights are the ratios of corresponding subsample size to the entire sample size. Table 3 presents the APSI results for nested theories in the dictator game. The second column presents APSI computed using the Bronars power and the third one presents APSI computed using the bootstrap power. First, we look at the results for Fisman, Kariv and Markovits (27) dataset. APSI for inequality aversion is at least as large as one for increasing benevolence. This result holds for both ways used to compute the power. Results for Porter and Adams (26) dataset look rather mixed. APSI is higher for increasing benevolence if Bronars power is used and APSI is higher for inequality aversion is higher if bootstrap power is used. Hence, inequality aversion is a rather prevalent theory, but it is not obviously dominant in terms of explaining 6 However, as a caveat here, note that power of test for increasing benevolence is significantly lower than the one for inequality aversion. Moreover, in Porter and Adams (26) experiment, three participants gave more to their counterparts than they kept for themselves in all budgets. Such altruistic behavior is maximally inconsistent with inequality aversion, implying an HMI of zero (theoretically HMI for other theories starts with ). We exclude these three subjects from the mixed type analysis without significantly affecting the results (see detailed results in Supplementary Materials).

23 REVEALED SOCIAL PREFERENCES 23 Bronars Bootstrap Fisman, Kariv and Markovits (27) data Inequality Aversion.55.3 Increasing Benevolence.3.3 Porter and Adams (26) data Inequality Aversion Increasing Benevolence.74.5 Table 3. APSI for Nested Theories in Dictator Game the observed behavior. Therefore, it is of particular importance to take both theories into account to explain the observed behavior. 3.2 Ultimatum Game We use data from Castillo, Cross and Freer (27) who conduct an experiment with total of 23 participants (students from Georgetown and Texas A&M universities). Every subject had to make the accept/reject decision over 3 alternatives drawn from 9 different linear budgets, that adds up to 7 binary choices from every subject. See Castillo, Cross and Freer (27) for the more detailed description of the design and the data as well as the evidence that subjects are consistent with other-regarding preferences. 7 Next, we present the results on testing inequality aversion in differences and inequality aversion in shares Consistency Results About 4-45% of the sample are consistent with either versions of inequality aversion without allowing for any decision making error. Results of the analysis from allowing for error of 5% are presented in the Table 4 (composition of the Table similar to those for dictator game). About 87% of subjects are consistent with inequality aversion in differences and 82% of subjects are consistent with inequality aversion of shares. In addition to Bronars and bootstrap power we also conduct the cutoff power analysis. Given the specifics of the budget sets we use the cutoff rules Idea behind is that responder uses cutoff rule everything below cutoff is rejected and everything above cutoff is accepted. This cutoff is determined randomly according to uniform distribution (Bronars cutoff) and empirical distribution functions (Bootstrap cutoff). Finally, 7 Castillo, Cross and Freer (27) conducted two sets of sessions, we report the results from them together. Let us note that choices are quite different between the populations, while the consistency with other-regarding preferences is quite uniform for both samples.